Question: Find the minimum value of
\[\sin^4 x + \frac{3}{2} \cos^4 x,\]as $x$ varies over all real numbers.
Explanation: By Cauchy-Schwarz,
\[\left( 1 + \frac{2}{3} \right) \left( \sin^4 x + \frac{3}{2} \cos^4 x \right) \ge (\sin^2 x + \cos^2 x)^2 = 1,\]so
\[\sin^4 x + \frac{3}{2} \cos^4 x \ge \frac{3}{5}.\]Equality occurs when
\[\sin^4 x = \frac{9}{4} \cos^4 x,\]or $\tan^4 x = \frac{9}{4}.$  Thus, equality occurs for $x = \arctan \sqrt{\frac{3}{2}}.$  Hence, the minimum value is $\boxed{\frac{3}{5}}.$